Diamagnetic Persistent Current in Diffusive Normal-Metal Rings E

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2-19-2001 Diamagnetic Persistent Current in Diffusive Normal-Metal Rings E. M.Q. Jariwala

P. Mohanty

M. B. Ketchen

Richard A. Webb University of South Carolina - Columbia, [email protected]

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Publication Info Published in Physical Review Letters, Volume 86, Issue 8, 2001, pages 1594-1597. Jariwala, E.M.Q., Mohanty, P., Ketchen, M.B., and Webb, R.A. (2001). Diamagnetic Persistent Current in Diffusive Normal-Metal Rings. Physical Review Letters, 86(8), 1594-1597. doi: 10.1103/PhysRevLett.86.1594 © 2001 The American Physical Society.

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Diamagnetic Persistent Current in Diffusive Normal-Metal Rings

E. M. Q. Jariwala,1 P. Mohanty,1,* M. B. Ketchen,2 and R. A. Webb1 1Center for Superconductivity Research, Department of Physics, University of Maryland, College Park, Maryland 20742 2IBM T. J. Watson Research Center, Yorktown Heights, New York 10598 (Received 9 December 1999) We have measured a diamagnetic persistent current with ﬂux periodicities of both h͞e and h͞2e in an array of thirty diffusive mesoscopic gold rings. At the lowest temperatures, the magnitudes of the currents per ring corresponding to the h͞e- and h͞2e-periodic responses are both comparable to the Thouless energy Ec ϵ h¯͞tD, where tD is the diffusion time. Taken in conjunction with earlier experiments, our results strongly challenge the conventional theories of persistent current. We consider a new approach associated with the saturation of the phase coherence time tw.

DOI: 10.1103/PhysRevLett.86.1594 PACS numbers: 73.23.Ra, 05.40.–a, 73.50.Mx p ͞ The phase of the electron wave function lies at the to e2 T T0 , and compare the different characteristic tem- heart of many mesoscopic phenomena. Consider a phase- peratures T0 to Ec. The sum of all the experimental results coherent ring of length L: the presence of an Aharonov- prompts a discussion of new calculations based on a direct Bohm flux F introduces a phase factor into the boundary connection to the saturation of tw at low temperatures. ͞ condition C͑x 1 L͒ ෇ C͑x͒e2piF F0 , such that all ther- Figure 1(a) displays a section of our array of 30 Au modynamic quantities are oscillatory in the applied flux, rings, fabricated using e-beam lithography on top of one with a period F0 ෇ h͞e. To minimize its free energy F, arm of the counterwound Nb pickup coil of a dc SQUID. this isolated ring will support an equilibrium persistent cur- A 150-nm-thick layer of SiO separates the coils from the rent I ෇ 2≠F͞≠F, even in the presence of disorder [1]. Au rings. We compute that the mutual inductance of each Moreover, for an ensemble of such rings, the fundamen- ring to the coil is 2.7 pH and uniform to within 5%. The tal harmonic of this current (with flux periodicity h͞e) gradiometer configuration of the two counterwound coils is strongly suppressed, whereas the next harmonic (with is clearly displayed in Fig. 1(b) together with the pair of flux periodicity h͞2e) survives due to the contribution of on-chip Nb field coils. Operating the dc SQUID in a flux- time-reversed paths of the electron [2]. locked loop mode, we measure a typical flux noise of Nanoampere-sized h͞e-orh͞2e-periodic currents have been separately observed in a few experiments: the h͞e current has been measured in single Au rings [3] and in a GaAs-AlGaAs ring [4], while the h͞2e current has been measured in multiring arrays of 107 Cu rings [5] and 105 GaAs-AlGaAs rings [6], where the h͞e response was not observable. Except for the case of the nearly ballistic GaAs ring [4], all the currents measured are remarkably 10–100 times larger than theoretically expected [7–9]. The diamagnetic sign of the average h͞2e response at high fre- quencies in Ref. [6] also contrasts with most predictions. Disorder plays a strong role in these results, emphasizing the Thouless scale of electron diffusion Ec. In this Letter, we report the first experiment on a single sample to measure the sign, amplitude, and temperature dependence for both the h͞e and h͞2e components of the persistent current. For our array of 30 diffusive Au rings, we determine the sign of both currents to be diamagnetic near zero field. At our lowest temperature of 5.5 mK, the amplitudes of the total h͞e and h͞2e currents are ͗I1T ͘ ෇ 1.92 nA and ͗I2T ͘ ෇ 1.98 nA, respectively. For comparison to previous experiments, wep present our results for the currents per ring as ͗I1͘ ෇ ͗I1T ͘͞ 30 ෇ ෇ ͞ ͗ ͘ ෇ ͗ ͘͞ ෇ ෇ 0.35 nA 2.3eEc h¯ and I2 I2T 30 0.066 nA FIG. 1. (a) Electron micrograph showing the Au rings, as well 0.44eEc͞h¯. We fit the temperature dependence of both the as the pickup and field coils of the SQUID. (b) Larger view ͞ h͞e and h͞2e currents to the form e2T T0 [8], as well as illustrating the entire gradiometer design of the SQUID.

1594 0031-9007͞01͞86(8)͞1594(4)$15.00 © 2001 The American Physical Society VOLUME 86, NUMBER 8 PHYSICAL REVIEW LETTERS 19FEBRUARY 2001 p ϳ3mFS͞ Hz down to 2 Hz at low temperatures. Here FS denotes the superconducting flux quantum. Each ring has a perimeter of length L ෇ 8.0 6 0.2 mm, with a line thickness t ෇ 60 6 2 nm and linewidth w ෇ 120 6 20 nm. We estimate Rᮀ ෇ 0.15 V͞ᮀ and a 2 diffusion constant of D ෇ yFle͞2 ෇ 0.06 m ͞s, where the elastic mean free path le Ӎ 87 nm. These parametersp as well as the phase coherence length Lw ϵ Dtw are measured separately in a 207 mm long meander with the same line thickness and line width as the rings. By using a standard weak localization measurement of the magnetore- sistance, we determined that Lw ഠ 16 mm for this control sample and is temperature independent below 500 mK 2 [10]. The Thouless energy Ec ϵ h¯͞tD ෇ hD¯ ͞L is 7.3 mK; the mean energy level spacing is given by D ϵ 1͞2N0V ෇ 19 mK, where N0 is the density of states and V is the volume of each ring. We measure the response of the rings to an applied magnetic field as a change in the average magnetic mo- ment coupled to the SQUID pickup coil. To distinguish the flux-periodic current in the rings from the background magnetization, we modulate the field at a small frequency f and measure separately the first, second, and third har- monics at f, 2f, and 3f, respectively. The amplitudes of the periodic signals measured are related to the ac modulation amplitude by a Bessel function, allowing us to maximize the signal in a particular harmonic for either ͞ ͞ FIG. 2. Magnetic response of 30 Au rings from a run focusing the h e or the h 2e flux-periodic response [11]. The on h͞2e oscillations. The dashed line in (a) represents raw response is independent of the measurement frequency f data. The solid line represents the h͞2e-periodic contribution, below 12 Hz, as also seen previously [3]. taken from the h͞2e window in the power spectrum in (b). In the following, we discuss a typical data set, shown The autocorrelation shown in (c) also demonstrates the h͞2e- in Fig. 2, which focuses on the h͞2e oscillations in the periodic signal. second harmonic of the magnetization using a 2.26 G ac modulation. As in most of our runs, we apply this ac field tion is consistent withp the corresponding weak-localization at f ෇ 2.02 Hz, which is superimposed on a dc field. The field scale of Hc ෇ 3¯h͞ewL. dc field is swept slowly over 620 G at 3.6 G͞h. The data A similar decaying envelope is seen in the h͞e-periodic is averaged in 0.08 G bins for over 80 s per bin, resulting response, as shown in Fig. 3(a). Here we use a 5.38 G ac in a noise floor of ϳ0.3mFS for each bin. In Fig. 2(a) modulation to maximize the h͞e oscillations in the third we display the raw data (dashes) of the second harmonic harmonic at 6.06 Hz. The solid line in Fig. 3(a) represents component at frequency 4.04 Hz taken at the base tem- the h͞e-periodic contribution to the total signal. Note that perature of 5.5 mK, along with the h͞2e-periodic contri- the leftmost peak shown in the power spectrum in both bution (solid line) represented by the signal in the h͞2e Figs. 2(b) and 3(b) results from the finite field range of window of the power spectrum shown in Fig. 2(b). The our data with a background subtracted. The size of our width of these windows corresponding to h͞e-orh͞2e- rings has been chosen to avoid confusion of the h͞e-or periodic currents reflect the finite linewidth of the rings; h͞2e-periodic signals with this lower-frequency response. furthermore, they are broadened to account for the effect The data in Figs. 2(a) and 3(a) also illustrate the dia- of the intrinsic width of the transform due to the finite field magnetic sign of the response near zero field. This de- range of the data. The autocorrelation function of the raw termination depends inherently on the exact wiring of the data is displayed in Fig. 2(c), clearly identifying an oscil- SQUID chip and electronics, but is confirmed empirically latory function in flux corresponding to an h͞2e-periodic by comparing an applied phase-modulated magnetic field response from the rings. Similar measurements maximiz- with the sign of the SQUID response in the higher harmon- ing the h͞2e response in the third harmonic at 6.06 Hz ics. Both the h͞2e and the h͞e responses are consistently yield similar results. One previously unreported observa- diamagnetic for all ac field drives used, at all temperatures tion is that the amplitude of the periodic response appears investigated, and on at least three different coldowns. 2H͞H to decrease exponentially as e c with Hc ഠ 6G.If Figure 4 displays the temperature dependence of the this decay is due to the dephasing effect of magnetic field total h͞2e and h͞e currents up to 150 mK. The values penetration into the arms of the ring [12], then our observa- for current are calculated from the flux response using the 1595 VOLUME 86, NUMBER 8 PHYSICAL REVIEW LETTERS 19FEBRUARY 2001

FIG. 4. Temperature dependences of the total (a) h͞2e and (b) h͞e currents calculated from the net magnetic response of the 30 Au rings. The solid lines are ﬁts to the form exp͑2T͞T0͒.

of 0.77eEc͞h¯ measured in a much larger ensemble of diffusive rings [5]. Both of these results are comparable to the sample-specific typical current Ityp predicted for a single ring (which should vanish in an array of rings), but are nearly 2 orders of magnitude larger than the expected value of ϳeD͞h¯ for the ensemble-averaged h͞2e current [7,9]. FIG. 3. Magnetic response of 30 Au rings from a run focus- ͞ Electron-electron interactions yield an h͞2e-periodic aver- ing on h e oscillations. The dashed line in (a) represents raw ͞ data. The solid line represents the h͞e-periodic contribution, age current within an order of magnitude of eEc h¯, but the taken from the h͞e window in the power spectrum in (b). The sign is required to be paramagnetic [8]. Substituting an at- autocorrelation is shown in (c). tractive phonon-mediated interaction finally yields the correct diamagnetic sign but reduces the amplitude in Ref. [8] mutual inductance and the appropriate Bessel function fac- by 1 order of magnitude [13]. For the h͞2e persistent cur- tor [11]. The solid lines drawn through the data are fits to ϳ ͞ ͞ rent, the combined result of the eEc h¯ magnitude (also ෇ 2T T0 ෇ I I0e and we find T0 89 and 166 mK for the in Refs. [5,6]) and the diamagnetic sign (also in Ref. [6]) h͞2e and h͞e responses, respectively. Our data canp also disagrees with all conventional theories. ͞ be fit over this limited temperature range to I ~ e2 T T0 For the h͞e response, the diamagnetic sign and the with T0 ෇ 25 mK for the h͞2e signal and 50 mK for the magnitude also present a challenge. The typical h͞e h͞e signal. Theoretically it is expected that T0 ~ Ec for current in any one ring is sample specific but of order typ both functional forms [7,8], but the comparison of our T0 I ϳ eEc͞h¯ [7]; a previous experiment detected para- with previous experiments suggests that this may not be magnetic h͞e currents almost 100 times larger, though, in the correct energy scale. For example, our T0 for the h͞2e two different rings [3]. For our experiment on an array signal is quite similar to that obtained in the Cu array ex- of N ෇ 30 rings,p we measure an h͞e current per ring ͗ ͘ ෇ ͗ ͘͞ ෇ periment [5], though their Ec was 3.3 times larger than of I1 I1T p30 0.35 nA at 5.5 mK, where we ours. A similar discrepancy arises in comparing the tem- have divided by N to account for the random sign of perature dependence of the h͞e signal in a single ring [3] the typical current. Corresponding to ͗I1͘ ෇ 2.3eEc͞h¯, to our present result. At least for our data, we can say that our result is much closer to the value Ityp predicted for a the h͞2e response is more strongly temperature dependent, single ring. Alternatively, if the diamagnetic h͞e signal with T0 about half that for the h͞e response given either from our array comes from the contribution of currents all exponential functional form. with the same sign,p then dividing our total h͞e response typ The sign of the persistent current, along with its magni- by N instead of N yields a value ͗I1͘ & I for the tude at T ෇ 0, is an integral part of most theories for the mean current per ring, but over 100 times larger than the h͞2e or the h͞e response. At our lowest T ෇ 5.5 mK, we more appropriate ensemble-averaged calculation of eD͞h¯ measure a diamagnetic h͞2e current of ͗I2͘ ෇ ͗I2T ͘͞30 ෇ [9]. Regardless of this question over the exact distribution 0.066 nA, dividing the total response by the number of of signs and amplitudes, the presence of a large h͞e signal rings to obtain the average current per ring. Correspond- in our array suggests that N ෇ 30 rings are not enough for ing to ͗I2͘ ෇ 0.44eEc͞h¯, our result is similar to the value ensemble averaging, as in Refs. [5,6]. New experiments 1596 VOLUME 86, NUMBER 8 PHYSICAL REVIEW LETTERS 19FEBRUARY 2001 on different arrays are needed here to clarify the dis- and T. R. Stevenson for discussions. This work is sup- tinction between the ensemble average and the typical ported by the NSF under Contract No. DMR 97-30577 currents in a finite array of rings. and by the ARO under Contract No. DAAG 55-97-10330. The significant gap between the h͞2e persistent current experiments and theory [14] suggests we look elsewhere for a solution. It is known that ac noise in a mesoscopic conductor can produce both a rectified dc current [15] and decoherence of the electron wave function [12]. Refer- *Present address: Dept. of Phys., Cond. Mat. Div. 114-36, ence [16] predicts that in a closed loop this noise induced California Institute of Technology, Pasadena, California 2 diamagnetic dc current is on the order of e͞tD ෇ eD͞L , 91125. within a factor of 2 of the experimental h͞2e persistent [1] N. Byers and C. N. Yang, Phys. Rev. Lett. 7, 46 (1961); current values. The high-frequency noise here can be from F. Bloch, Phys. Rev. B 2, 109 (1970); M. Buttiker, Y. Imry, external incoherent sources or can arise internally, e.g. and R. Landauer, Phys. Lett. 96A, 365 (1983). [2] H. Bouchiat and G. Montambaux, J. Phys. (Paris) 50, 2695 from two level systems [17], 2-channel Kondo impurities (1989); G. Montambaux et al., Phys. Rev. B 42, 7647 [18], or nuclear spins. It was recently suggested [19,20] (1990). that this same ac noise may also explain the theoretically [3] V. Chandrasekhar et al., Phys. Rev. Lett. 67, 3578 (1991). unexpected saturation of the dephasing time tw at low tem- [4] D. Mailly, C. Chapelier, and A. Benoit, Phys. Rev. Lett. peratures [10]. In fact, Ref. [20] gives a universal relation- 70, 2020 (1993). ship ͗I2͘ ෇ Ce͞t0 (where the constant C of order 1 also [5] L. P. Levy et al., Phys. Rev. Lett. 64, 2074 (1990); contains the sign), connecting the ensemble-averaged h͞2e L. P. Levy, Physica (Amsterdam) 169B, 245 (1991). current ͗I2͘ to the dephasing time t0 at T ෇ 0 via the same [6] B. Reulet et al., Phys. Rev. Lett. 75, 124 (1995). [7] H. F. Cheung, E. K. Riedel, and Y. Gefen, Phys. Rev. Lett. nonequilibrium noise. Estimating t0 ഠ 4.2 ns [10] and ac- counting for strong spin-orbit scattering in our gold rings, 62, 587 (1989). this formula yields a value of 0.025 nA but paramagnetic [8] V. Ambegaokar and U. Eckern, Phys. Rev. Lett. 65, 381 (1990); 67, 3192 (1991); U. Eckern, Z. Phys. B 82, 393 in sign, as compared to our diamagnetic measurement of (1991). 0.066 nA per ring. For larger arrays of rings with weaker [9] A. Schmid, Phys. Rev. Lett. 66, 80 (1991); F. von Oppen spin-orbit scattering, this new idea appears to also give the and E. K. Riedel, ibid. 66, 84 (1991); B. L. Altshuler, correct h͞2e amplitude based on t0 [5,6], as well as the ob- Y. Gefen, and Y. Imry, ibid. 66, 88 (1991). served diamagnetic sign [6]. We add that energy averaging [10] P. Mohanty, E. M. Q. Jariwala, and R. A. Webb, Phys. Rev. effects may explain why the persistent current continues to Lett. 78, 3366 (1997). Our control sample is “Au-2.” grow below the temperature at which tw saturates. We also [11] The exact Bessel function relation is described in Ref. [5]. believe that the origin of this ac noise must be internal to [12] B. L. Altshuler and A. G. Aronov, in Electron-Electron In- the sample because recent dephasing measurements in the teractions in Disordered Systems, edited by A. L. Efros and presence of external ac noise [21] cannot explain the mag- M. Pollak (Elsevier, Amsterdam, 1985); S. Chakravarty and nitude or saturation behavior of t in Au wires. A. Schmid, Phys. Rep. 140, 193 (1986). w [13] V. Ambegaokar and U. Eckern, Europhys. Lett. 13, 733 In summary, we have presented the first temperature ͞ ͞ (1990). dependent measurements of both the h 2e and h e com- [14] U. Eckern and P. Schwab, Adv. Phys. 44, 387 (1995). ponents of the persistent current from an array of 30 dif- [15] V.I. Fal’ko and D. E. Khmelnitskii, Zh. Eksp. Teor. Fiz. fusive Au rings. The magnitude of our h͞2e response 95, 328 (1989) [Sov. Phys. JETP 68, 186 (1989)]. corresponds to an average current per ring of ϳeEc͞h¯,in [16] V.E. Kravtsov and V.I. Yudson, Phys. Rev. Lett. 70, 210 agreement with previous experiments in multiple-ring sys- (1993). tems. The magnitude of our h͞e current is also found to [17] Y. Imry, H. Fukuyama, and P. Schwab, Europhys. Lett. 47, be ϳeEc͞h¯.Wefind a diamagnetic response for both the 608 (1999). h͞e and h͞2e signals. These results, including the origin [18] A. Zawadowski, J. von Delft, and D. C. Ralph, Phys. Rev. Lett. 83, 2632 (1999). of T0 in the temperature dependence, cannot be explained within the framework of existing theories. The new sug- [19] P. Mohanty, in special issue on The Proceedings of Localization-1999, Hamburg, Germany, edited by gestion that the h͞2e current and the saturation of t re- w M. Schreiber [Ann. Phys. (Leipzig) 8, 549 (1999)]. sult from the same internal ac noise appears promising but [20] V.E. Kravtsov and B. L. Altshuler, Phys. Rev. Lett. 84, needs to be explored more carefully. 3394 (2000). We thank B. L. Altshuler, V. Ambegaokar, H. Bouchiat, [21] R. A. Webb et al., in Quantum Coherence and Deco- Y. Gefen, Y. Imry, V.E. Kravtsov, D. Mailly, herence–Proceedings of ISQM Tokyo 1998, edited by G. Montambaux, P. Schwab, J. Schwarz, R. Smith, K. Fujikawa and Y.A. Ono (Elsevier, Amsterdam, 1999).

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